1. COMPOSITION OF FUNCTIONS
Example 1
If and , find
COMPOSITION OF FUNCTIONS

2. DECOMPOSITION OF FUNCTIONS
Example 2
If each function below represents , define and
DECOMPOSITION OF FUNCTIONS

3. The Chain Rule
If y = f(u) is a differentiable function of u and u = g(x) is a differentiable function of x, then y = f(g(x)) is a differentiable function of x, and
Other ways to write the Rule:

4. Instructions for The Chain Rule
For , to find :
Decompose the function
Differentiate the MOTHER FUNCTION
Differentiate the COMPOSED FUNCTION
Multiply the resultant derivatives
Substitute for u and Simplify
Make sure each function can be differentiated.

5. Example 1 Find if and . Define f and u:
Find the derivative of f and u:

6. Example 2 Differentiate . Define f and u:
Find the derivative of f and u:

7. Example 3 If f and g are differentiable, , , and ; find .
Define h and u:
Find the derivative of h and u:

8. Example 4
Find if
Define f and u:
Find the derivative of f and u:

9. Example 5 Differentiate . Define f and u:
Find the derivative of f and u: OR

10. Now try the Chain Rule in combination with all of our other rules.

11. Example 1 Differentiate . Use the old derivative rules
Chain Rule Twice

12. Example 2 Find the derivative of the function . Quotient Rule
Chain Rule

13. Example 3
Differentiate
Chain Rule Twice

14. Example 4
Differentiate
Chain Rule
Chain Rule Again

15. Example 5 Find an equation of the tangent line to at .
Find the Derivative
Evaluate the Derivative at x = π
Find the equation of the line